problem on A.P. arithmetic progression

find the sum of all even natural numbers between 2 and 200 (both inclusive ) excluding the multiples of 3.

multiple of both 2 and 3 means multiple of 6 (lcm)

formulae on progressions A.P. G.P H.P. etc

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orthogonal trajectory of y(1+x ² ) = Cx

find the orthogonal trajectory of y(1+x ² ) = Cx
first form the differential equation of the given family
replace (dy/dx) with -1 / (dy/dx) to get the differential equation of the orthogonal trajectory
and solve the resulting equation

another example:orthogonal trajectory of y = (k/x)
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area bounded by |x| + |y| = 20

|x| + |y| = 20
corresponds to
x + y=20
x - y=20
-x + y=20
-x -y =20
the graph is shown below


required area = 4*area of right triangle

= 4 * (1/2) *20 * 20 =800 sq. units
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integral of (5x^3 + 8x^2 + 36x + 36) / (x^4 + 9x^2)

integral of (5x^3 + 8x^2 + 36x + 36) / (x^4 + 9x^2)
first do the partial fraction decomposition
refer partial fraction decomposition
then split the (x+4) /( x² +9) into two terms
the first one can be manipulated into u' / u form


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limits for the correlation coefficient

find limits for the correlation coefficient

show that -1 <= r <=1 or |r| <=1

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equation of tangent

find the equation for the tangent to y = x^4 at the point where x=2
first substitute x=2 into the given equation to find y to get the point (x1,y1)
at which the tangent is required.
find dy/dx and evaluate it at (x1,y1) to get the slope,m of the tangent
then use (y-y1) = m(x-x1) to get the equation of the tangent

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software resources

portable apps are software that can be carried around on usb sticks, and will work without further installation onto the computer and can hence be run on any computer from which the user can access his portable data storage . Examples like portable versions of GNUcash ( accounting/financial management software) , firefox browser, bonkenc (for extracting music for cd's and changing them into formats like mp3), abiword portable version for wordprocessing , virtualdub portable version for video processing and capture etc
available at portableapps.com

some software for image resizing,adding text,watermark,crop, batch rename, screen capture are available at faststone.org some of the software are freeware and some are shareware
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second derivative at a given point with implicit differentiation

if x² +y² = 25, find the value of d²y / dx² at (4,3)
differentiate once , extract the value of dy/dx
find the value at (4,3)
differentitate the expression for dy/dx once more
to get d²y / dx²
it might contain dy/dx
substitute (4,3) and the value obtained earlier for dy/dx

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some nice unconventional foodstuffs

seaweed :some varieties might be rich in calcium, magnesium

pumpkin seeds might be a source of vitamins, protein, zinc, magnesium and manganese

unheated, unprocessed raw honey might contain calcium,iron, magnesium,copper, antioxidants etc


cocoa beans might contain some antioxidants and also give some cardiovascular benefits

comic stuff

how to keep the ignorant busy for a short time

ordinary english to chat language translator

some more resources for weight loss, exercise etc

to find the calorie content of some common foods, try
calorieconnect.com

some useful calculators (not sure of their accuracy ) for bmi, fat content etc can be found at
healthstatus.com

a calorie calculator
caloriesperhour.com

videos of a collection of exercises
uwlax.edu


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